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Distributions pp 153-176 | Cite as

Fractional Integration and Differentiation

  • J. J. Duistermaat
  • J. A. C. Kolk
Chapter
Part of the Cornerstones book series (COR)

Abstract

In this chapter we deal with “complex powers of the operator \(\frac{d}{{dx}},\)” a concept already found in the posthumous works of Riemann and elaborated by Marcel Riesz in the 1930s and 1940s. The relevant article [18] is lengthy, but with the help of a little distribution theory and complex analysis, all results can readily be proved.We will also deal with Riesz’s treatment of the wave operator \(\Box = \partial _t^2 -\triangle_{x}\) in arbitrary dimension; thus we will obtain, among other things, a fundamental solution of \(\Box.\)

Keywords

Cauchy Problem Fundamental Solution Analytic Continuation Radon Measure Fractional Integration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Mathematical InstituteUtrecht UniversityUtrechtThe Netherlands

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